Assertion- Reason Questions
Q1: Assertion (A): A trapezium with 3 equal sides and one side double the equal side can be divided into three equilateral triangles of equal area.
Reason (R): Area of a trapezium =\(\frac{1}{2}\) × (sum of parallel sides) × distance between them.
Explanation:
– The given trapezium with 3 equal sides and the fourth side double the length of one equal side forms a special shape that can be divided into three equilateral triangles of equal area. So, Assertion (A) is true.
– The formula for the area of a trapezium given in Reason (R) is the correct standard formula.
– However, Reason (R) gives the general formula for trapezium area but does not explain why the trapezium in Assertion (A) can be divided into three equilateral triangles.
– Therefore, while both Assertion and Reason are true, Reason (R) is not the correct explanation for Assertion (A).
Answer: b. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
Q2: Assertion (A): If the perimeter of a semi-circle is 18 cm, then its radius is 3.5 cm.
Reason (R): The perimeter of a semi-circle of radius r is r (π + 2).
Step 1: Formula for perimeter of semi-circle:
\[
P = \pi r + 2r = r(\pi + 2)
\]Step 2: Given perimeter \(P = 18\) cm, find radius \(r\):
\[
18 = r(\pi + 2)
\]
Taking \(\pi = 3.14\),
\[
18 = r(3.14 + 2) = r \times 5.14 \\
r = \frac{18}{5.14} \approx 3.5 \, cm
\]Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).
Answer: a. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Q3: Assertion (A): The side of a rhombus is 10 cm. If its one diagonal is 12 cm long, then its area is 24 cm².
Reason (R): Area of a rhombus is given by product of its diagonals.
Step 1: Area formula for a rhombus:
The correct formula is:
\[
\text{Area} = \frac{1}{2} \times d_1 \times d_2
\]
where \(d_1\) and \(d_2\) are the lengths of the diagonals.
Step 2: Given side \(s = 10 \, cm\), one diagonal \(d_1 = 12 \, cm\), find the other diagonal \(d_2\):
Since the diagonals bisect each other at right angles,
\[
s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \\
10^2 = \left(\frac{12}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \\
100 = 6^2 + \left(\frac{d_2}{2}\right)^2 = 36 + \frac{d_2^2}{4} \\
\frac{d_2^2}{4} = 100 – 36 = 64 \Rightarrow d_2^2 = 256 \Rightarrow d_2 = 16 \, cm
\]Step 3: Calculate area:
\[
\text{Area} = \frac{1}{2} \times 12 \times 16 = \frac{1}{2} \times 192 = 96 \, cm^2
\]The assertion says area is 24 cm², which is false.
The Reason states the area is the product of diagonals (without half), which is incorrect.
Answer: Assertion (A) and Reason (R) are false.
Leave a Comment